“Rich kids can always get Algebra or Calculus”
On Substack, Bari Weiss sums up the week's craziness, including California's decision to trap all 8th graders in pre-algebra in the interest of the usual murky goals. She quotes Freddie DeBoer's observation that families with extra cash will just hire tutors, so this equity-inclusion push will consign only the smart poor kids to the needlessly crummy education track. The truth is, though, that these days any kids can get decent algebra or calculus instruction with or without a tutor. Even the poor kids have some kind of access to the internet, where the educational resources are nearly endless. Any kid that was likely to be able to pick up calculus from high school lectures will be able to get it from internet lectures, if not from a book. You don't even have to be Isaac Newton, who, when he found he lacked this essential tool, simply created it during one of the Western World's more famous lockdowns.On the other hand, the way things are going, will there still be colleges where you can go anywhere with higher math? I'd love to see aspiring young workers skip the whole thing, learn the math on their own, and get jobs in STEM industry, minus the political indoctrination.The lockdown link by the way, is a windy attempt to explain why no one should feel bad about not doing world-changing work during lockdown because privilege or something. The problem certainly isn't just that we lack a one-in-a-billion talent! Probably any of us could have pulled it off if we had a Universal Basic Income and some domestic servants.
I taught myself most of the math I learned, though the teachers were useful at odd points. When I got to take an advanced math course between 11th-12th grades, however, I also learned a great deal from excited classmates. The professor, borrowed from a fancy prep school, was also great. I think smart poor kids like me can mostly do fine as long as they can get some usable material and we aren't actively beating them. Yet there may be a value-added from being with other good students that is missed. The history of great mathematicians is a lot of autodidacts, but that may not be relevant to what schools are doing.
ReplyDeleteIt is a long-time favorite topic of mine, as you know. Here is one entry: https://assistantvillageidiot.blogspot.com/2013/12/math-should-be-taught-like-literature.html
One overlooked aspect of highschool math instruction is that we are looking to measure the ability to do abstract thought, even if the actual math used is forgotten a few years later.
I have a good deal of sympathy for your idea of moving the math and science kids to STEM fields and out of academia more quickly. However, as Silicon Valley is now nearly as woke as Berkeley, that may not help either. Also, math, especially statistical analysis, is being used in more and more fields these days. The trend is cross-disciplinary,and that's a good thing.
Usually.
Online learning is not always workable, as we have recently had occasion to study in depth. Some students responded well, but for very many 2020-1 was a lost year academically.
ReplyDelete@ Grim - yes, and I think that even though all research will be initiated to prove that hybrid and online schooling was either just fine or horribly destructive*, we will find that while most of it is corrected for pretty rapidly, there will be specific slots where students did worse because of it. My first nominations would be those late HS students trying to demonstrate competence in high-abstraction, and maybe the youngest children in their social development. Those might turn out to be measurable.
ReplyDelete*education research on all sides is about the worst that is out there. It's mostly crap.
The truth is, though, that these days any kids can get decent algebra or calculus instruction with or without a tutor.
ReplyDeleteWhat some kids don't realize is that often the textbook can be a good teacher. My 9th grade math teacher was no dummy- a Phi Beta Kappa graduate of U Michigan- but her classroom management skills were abysmal. As I couldn't learn any math from her chaotic classrooms, I taught myself from the textbook. The text came from New Math program developed at the University of Illinois. From the start it emphasized writing proofs.
Before my 9th grade math class, I was good in math, but indifferent to it. As a result of my 9th grade math class, I loved math. It also became my best subject. All this from learning from the textbook.
I find this talk about tutors rather amusing, as I didn't know of any peers ever using tutors.
I'd add that my other math teachers in high school were good. One, who taught me in 10th and 12th grades, was an exceptionally good teacher. She had a knack for making it look easy.
ReplyDelete@ Gringo - we used a tutor for another reason with Son #4 who came from Romania and had very poor education before coming to us. It's often a rescue operation. Tutoring to teach at upper levels? I think that energy is put into test prep, which has a mild effect. (They make their money because kids naturally go up about 90 points on their SATs in a year whether they get test prep or not. The expensive companies take credit for that.)
ReplyDeleteWhatever happened to teaching geometry? For a couple of millennia "mastering Euclid" defined an educated person. Geometry provided a foundation for trigonometry, thence to surveying, orienteering, navigation, and civil engineering. While things aren't MOVING, calculus is extra.
ReplyDeleteAnyhow,
https://mathwithbaddrawings.com/2019/11/04/the-tilted-twin-and-other-delights/
Abraham Lincoln studied Geometry because he thought it would help teach him to develop logical arguments.
ReplyDelete(Although geometry was required for everybody in my high school, IIRC, and I'm not sure it led to any great outbreak of transferrable logical thinking skills)
It probably did when he studied it. The way algebra and geometry was taught changed after mathematicians worked out that they could both express the same information. It used to be taught in the Euclidean way, in which you began with axions and then reasoned from them. That’s still a helpful thing to learn to do, though we teach logic and deductive systems differently now.
ReplyDelete"While things aren't MOVING, calculus is extra"..ah, but things often *are moving...
ReplyDeleteThe late and very great blogger Lex remarked that he had not done terribly well at math in high school and the first two years of college:
"It was not until my junior year at the Naval Academy, when we started to do differential equations, that the light came on. Eureka! Drop a wrench from orbit, and over time it would accelerate at a determinable pace, up until the moment when it entered the atmosphere, where friction would impede the rate of acceleration at an increasingly greater rate (based on air density, interpolated over a changing altitude) and that wrench struck someone’s head at a certain velocity, that any of this applied in the real word. By then it was too late, I was too far gone, and an opportunity was lost."
There is a group at Marshall University (West Virginia) which has built some old-style mechanical differential analyzers and believes that they could be of great use in math teaching:
https://chicagoboyz.net/archives/57194.html
In my experience, I learned a lot from the textbooks without the teacher, but there were several points where the "aha" didn't kick in until somebody was actually waving his hands.
ReplyDeleteTeachers can make it easier, especially if they mastered the material themselves and know different ways to explain it. Not all meet those conditions.
I did poorly in math all through grade school, because I came through just as "drill-n-kill" was no longer being taught. I need that sort of work to lay a foundation. Otherwise I founder, because I am/was very slow learning abstract ideas. I squeaked through HS geometry, and went no farther. I need good teachers, and a strong foundation. The on-line videos and courses probably would not have helped me much (they don't now.)
ReplyDeleteHowever, once I was doing applied math in college (trig) and in aviation, I did very well. Among other things I discovered that I could design a bridge that would not fall down easily. I was invited to take college calculus, but declined. Without a solid foundation, I knew I would fail.
Now, I surprise my students by doing math in my head. I'm slow, and I have to visualize more complex things as if I were drawing them on a piece of paper, but I can do it.
LittleRed1
"It used to be taught in the Euclidean way, in which you began with axions and then reasoned from them."
ReplyDeleteThis is probably why I *loved* learning geometry while almost all my classmates hated it (and I hated all other math classes). It's a shame if that's no longer how it's taught. There would be a lot lost in dropping those techniques.
These days it’s taught as a kind of less-abstract form of algebra. Usually after algebra 1 and 2, once kids are familiar with how algebra works, you start showing them how those techniques can be used to describe and define shapes. This allows you to then go into functions and modeling, which allows you to mathematically describe other things than basic shapes. Then you go into pre calculus and then calculus, which allow you to mathematically describe Newtonian physical applications. Then you have what you need to grasp how to apply it all to physical objects moving in space under physical conditions— I.e., the math you need for engineering.
ReplyDeleteBut it does lose the Euclidean approach entirely; you have no need to ever mention him nor discuss his approach. And it also loses something of AVI’s approach to testing for increasing levels of abstract thought. Algebra one and two still seem very abstract, but starting with geometry you begin to learn practical applications for the earlier techniques. Eventually you can use math to design advanced technologies, but even before calculus you can design video games. (Or at calculus, if you want them to have realistic physics.)
I'm trying to remember from so many decades ago how I actually learned math. My father did tutor me at times, but I don't remember that being terribly successful for the most part. He was gifted with adult math nerds, not so much with getting back into the mindset of someone completely innocent of the basic concepts. I must have had some teachers with that talent, though it didn't become apparent until I was old enough to get into the accelerated classes. Certainly being among students who enjoyed it and had a knack for it helped.
ReplyDeleteHonestly, though, my dominant memory is of the explosive hit on the pleasure centers in my brain when systems began to make sense and it became possibly to solve problems. If it hadn't been such an overwhelming pleasure, who knows if I'd ever have bothered learning it. Almost nothing in my early memories compares with it. Does anyone? Am I lazier than most?
It seems to be that kids will learn anything that they have the horsepower to learn if they can be made to want it. Making them want it is the trick--either for the sake of its inherent pleasure, or as a tool to get something they badly want. Bailey White, for instance, found that it was a snap to teach kids to read if they were ravenous for the end of a story. The HBO show "The Wire" had a terrific sequence on teaching kids math so they could make money playing craps. I did it for the pleasure hit, the "aha" moment. It really is like a volcano going off in the head. Anyone addicted to crossword puzzles and the like will recognize the compulsion and the pleasure, the ecstatic "Ohhhhhh!"
I didn't do homework as a rule in grade school, except for long hours of happy work on math or physics problems once I got old enough for those. The teacher would get us started on a technique, and I did need that initial interaction, but then the rest was up to me.
It helps to have a teacher's undivided attention, so you can stop and ask questions when the message isn't coming across, but I found that video lectures of the Khan Academy work pretty well, too, since you can stop and start and replay, or try someone else's lecture, who may put the concepts in a slightly different way.
I didn't find learning math from books particularly easy. Ditto physics, though I highly recommend Richard Feynman's 3-volume CalTech set.
You were not nearly as lazy as I was Tex. I can assure you of that. If only I'd realized the pleasures of math success earlier in my youth, I might have been better at it.
ReplyDelete